86
2 Macroscopic Thermodynamics
from which we may deduce that for any irreversible process that takes place at
constant S and V , we have
δQ
= −dU > 0 .
(2.6.2b)
This result is more commonly included in a summary statement
dU ≤ 0 ,
(2.6.2c)
in which the equality holds for a reversible thermodynamic process and the
inequality holds for an irreversible thermodynamic process. The result (2.6.2c) tells
us that an irreversible change at constant S and V results in a decrease in the internal
energy, so that U thus plays the part of an indicator of irreversible processes for
changes that occur at constant S and V . The non-occurrence of irreversible process
when U remains constant for a closed system and the inevitability of spontaneous
irreversible processes occurring when U decreases for a closed system gives rise
to the terminology ‘thermodynamic potential’ for U in association with adiabatic
isochoric processes.
If we consider systems for which S and P , rather than S and V , are the relevant
system variables, then the appropriate thermodynamic potential will be the enthalpy,
H . Hence, irreversible processes at constant S and P will, because dH takes the
form
dH = T dS + V dP − δQ
,
(2.6.3a)
then lead to
δQ
= −dH > 0 ,
(2.6.3b)
so that irreversible processes will, in this case, be indicated by a decrease in the
enthalpy of the system. Similarly, we may consider systems for which T and V or
T and P are the relevant system variables. If the relevant system variables are T and
V , the appropriate thermodynamic potential is then the Helmholtz energy, A, from
which we find that
dA = −SdT − P dV − δQ
,
(2.6.4a)
and hence
δQ
= −dA > 0 ,
(2.6.4b)
or equivalently,
dA ≤ 0 ,
(2.6.4c)
2 Macroscopic Thermodynamics
from which we may deduce that for any irreversible process that takes place at
constant S and V , we have
δQ
= −dU > 0 .
(2.6.2b)
This result is more commonly included in a summary statement
dU ≤ 0 ,
(2.6.2c)
in which the equality holds for a reversible thermodynamic process and the
inequality holds for an irreversible thermodynamic process. The result (2.6.2c) tells
us that an irreversible change at constant S and V results in a decrease in the internal
energy, so that U thus plays the part of an indicator of irreversible processes for
changes that occur at constant S and V . The non-occurrence of irreversible process
when U remains constant for a closed system and the inevitability of spontaneous
irreversible processes occurring when U decreases for a closed system gives rise
to the terminology ‘thermodynamic potential’ for U in association with adiabatic
isochoric processes.
If we consider systems for which S and P , rather than S and V , are the relevant
system variables, then the appropriate thermodynamic potential will be the enthalpy,
H . Hence, irreversible processes at constant S and P will, because dH takes the
form
dH = T dS + V dP − δQ
,
(2.6.3a)
then lead to
δQ
= −dH > 0 ,
(2.6.3b)
so that irreversible processes will, in this case, be indicated by a decrease in the
enthalpy of the system. Similarly, we may consider systems for which T and V or
T and P are the relevant system variables. If the relevant system variables are T and
V , the appropriate thermodynamic potential is then the Helmholtz energy, A, from
which we find that
dA = −SdT − P dV − δQ
,
(2.6.4a)
and hence
δQ
= −dA > 0 ,
(2.6.4b)
or equivalently,
dA ≤ 0 ,
(2.6.4c)